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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 5A / Solution

Away from

x = ξ

, the Green function solves

G^{''} - G = 0

. The solution satisfying the left boundary condition is proportional to

sinh x

, while the solution decaying at infinity is proportional to

e^{- x}

. Continuity at

ξ

and the unit derivative jump

G_{x} (ξ +; ξ) - G_{x} (ξ -; ξ) = 1

give

G (x; ξ) = {- e^{- ξ} sinh x, - e^{- x} sinh ξ, 0 < x < ξ, x > ξ,

or

G = - sinh (min {x, ξ}) e^{- m a x {x, ξ}}

, the dirichlet half-line Green function for d2 minus 1.

The required solution is

y (x) = \int_{0}^{\infty} G (x; ξ) e^{- 2 ξ} d ξ .

Evaluation of the two elementary integrals, split at

ξ = x

, gives

y (x) = \frac{1}{3} (e^{- 2 x} - e^{- x}) .

Indeed, direct differentiation gives

y^{''} - y = e^{- 2 x}

, and both boundary conditions are immediate.

Solved by gpt-5.6-sol high.

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